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A note on the equilibrium M/G/1 queue length

Published online by Cambridge University Press:  14 July 2016

Gordon E. Willmot
Gordon E. Willmot*
Affiliation:
University of Waterloo
*
∗ Postal address: Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1.
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Abstract

This note concerns the distribution of the equilibrium M/G/1 queue length. A representation for the probability generating function is given which allows for an explicit finite sum representation of the associated probabilities. The radius of convergence of the probability generating function and an asymptotic formula for the right tail of the distribution also follow from this representation, as well as infinite divisibility of the queue-length distribution when the service distribution is infinitely divisible. Extension of these results to the bulk arrival case is straightforward.

Keywords

RENEWAL THEOREMCOMPOUND GEOMETRICMIXED POISSONINFINITE DIVISIBILITYBULK ARRIVALS
Type
Short Communications
Information
Journal of Applied Probability , Volume 25 , Issue 1 , March 1988 , pp. 228 - 231
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Research supported by the Natural Sciences and Engineering Council of Canada.

References

Chaudhry, M. and Templeton, J. (1983) A First Course in Bulk Queues. Wiley, New York.Google Scholar
Embrechts, P. (1983) A property of the generalized inverse Gaussian distribution with some applications. J. Appl. Prob. 20, 537–544.CrossRefGoogle Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications , Vol. 1, 3rd edn. Wiley, New York.Google Scholar
Maceda, E. (1948) On the compound and generalized Poisson distributions. Ann. Math. Statist. 19, 414–416.CrossRefGoogle Scholar
Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
Van Harn, K. (1978) Classifying Infinitely Divisible Distributions By Functional Equations. Math. Centre Tracts 103, Math. Centre, Amsterdam.Google Scholar
Van Hoorn, M. (1984) Algorithms and Approximations for Queueing Systems. CW1 Tract No. 8, CW1 Amsterdam.Google Scholar